Real Log Canonical Thresholds

• Real log canonical thresholds (RLCTs) are numerical invariants that define the largest exponent ensuring local integrability of |f|^{-c} near singularities.
• They are computed via resolution of singularities and Newton polyhedra, yielding explicit formulas and sharp bounds for various real analytic functions.
• RLCTs play a critical role in Bayesian statistics by determining learning coefficients that influence asymptotic behavior and model selection criteria.

A real log canonical threshold (RLCT), also referred to as the critical integrability index, is a fundamental numerical invariant associated with a real-analytic function, ideal, or pair at a point on a smooth real or complex manifold. It generalizes the notion of the log canonical threshold from algebraic geometry to the real-analytic and statistical settings and governs the convergence of integrals of the form $\int |f|^{-c}$ near the singular locus. RLCTs play a central role in singularity theory, birational geometry, and Bayesian statistics, where they appear as the “learning coefficient” controlling asymptotic learning rates in singular models.

1. Definition and Basic Properties

Let $f : (\mathbb{R}^n, 0) \to \mathbb{R}$ be a real analytic germ with $f(0) = 0$. The real log canonical threshold at 0 is defined as

$$\operatorname{lct}_0(f) = \sup\left\{ c > 0 \;\bigg|\; \text{there exists a neighborhood } U \ni 0 \text{ with } \int_{U} |f(x)|^{-c} dx < \infty \right\},$$

i.e., it is the largest $c$ for which $|f|^{-c}$ is locally integrable around the origin (Collins, 2017, Mustata, 2011). For a multi-ideal $(a_1^{e_1}\cdots a_r^{e_r})$ with real exponents on a smooth variety, the RLCT is the supremum $c$ such that the pair is log canonical at a given point, and can be computed as an infimum over certain divisorial valuations:

$$\operatorname{lct}_x(X, a^e) = \inf_E \frac{k_E+1}{v_E(a^e)},$$

where $E$ ranges over prime divisors above $X$ with center containing $x$, $k_E$ is the log discrepancy, and $v_E(a^e) = \sum e_j v_E(a_j)$ (Ishii, 30 Jun 2025).

The RLCT admits several basic properties:

• Monotonicity: If $|g| \leq C|f|$ near $x_0$, then $$\operatorname{lct}_{x_0}(g) \geq \operatorname{lct}_{x_0}(f)$$.
• Invariance under local real-analytic diffeomorphism: $$\operatorname{lct}_0(f \circ \varphi) = \operatorname{lct}_0(f)$$ if $\varphi$ fixes $0$.
• Semicontinuity: In smooth families, the RLCT is lower semicontinuous in parameters (Mustata, 2011).

For smooth monomial examples, $f(x) = x^m$ yields $\operatorname{lct}_0(f) = 1/m$ in one variable; for $f(x,y) = x^a y^b$ on $\mathbb{R}^2$, one has $\min\{1/a, 1/b\}$ (Collins, 2017, Mustata, 2011).

2. Resolution of Singularities and Computation Methods

The general computation of RLCTs for analytic or polynomial functions proceeds via resolution of singularities. Let $f \in \mathbb{R}[x_1, ..., x_n]$ and let $\sigma: U \to \mathbb{R}^n$ be a proper real-analytic map such that locally,

$$f \circ \sigma(u) = \eta(u) \prod_{i=1}^{n} u_i^{a_i}, \quad |\det D(\sigma)(u)| = \eta'(u) \prod_{i=1}^{n} u_i^{b_i},$$

with $\eta, \eta'$ real-analytic and nonvanishing. The RLCT is then

$$\lambda_R = \inf_{P \in U, a_i > 0} \frac{b_i + 1}{a_i},$$

and its multiplicity $m_R$ is the maximal number of indices $i$ for which this minimum is achieved at some chart point $P$ (Kosta et al., 20 Nov 2024).

For general pairs $(X, B)$, with $B$ an $\mathbb{R}$-divisor and $L$ an $\mathbb{R}$-Cartier divisor, the log canonical threshold is

$$\operatorname{lct}(X, B; L) = \inf_T \frac{a(T, X, B)}{\operatorname{mult}_T(\varphi^*L)},$$

where $a(T, X, B)$ is the log discrepancy of $T$ (Birkar, 2016).

In dimension two, an explicit method uses the Newton polyhedron associated to the Taylor expansion of $f$. The Newton polyhedron $N P(f)$ is the convex hull of $(p,q) + \mathbb{R}_{\geq 0}^2$ for all monomials $x^p y^q$ with $a_{p,q} \neq 0$. The Newton distance $d_{NP}(f)$ is the least $t$ such that $(t,t) \in NP(f)$. After an adapted real-analytic coordinate change, the RLCT is

$\operatorname{lct}_0(f) = 1 / d_{NP}(f)$

(Collins, 2017).

3. Structural Properties: ACC, Rationality, and Accumulation Points

The set of possible real log canonical thresholds in fixed dimension enjoys the Ascending Chain Condition (ACC), i.e., it contains no infinite strictly increasing sequence. In dimension two, all positive accumulation points of strictly decreasing sequences of thresholds are reciprocals of positive integers—that is, $1/m$ with $m \in \mathbb{N}$—matching those arising in dimension one (Collins, 2017, Ishii, 30 Jun 2025).

Tables: Key structural properties of RLCTs in various settings.

Setting	ACC Holds	Accumulation Points	Rationality (for $\mathbb{Q}$ exponents)
$\mathbb{R}$-analytic, $n=2$	Yes	$1/m$ $(m\in\mathbb{N})$	Conjectured
Characteristic $p$	Yes	Rational	Yes (Ishii, 30 Jun 2025)
Hyperplane arrangements	Yes	Explicit combinatorial	Yes if all multiplicities are integer

In positive characteristic, for multi-ideals with real exponents, the set of RLCTs is contained in that of characteristic zero, is discrete for fixed exponents, satisfies ACC, and has only rational accumulation points when exponents are rational (Ishii, 30 Jun 2025).

4. Illustrative Examples and Explicit Computations

• Plane cusp: $f(x, y) = y^2 - x^3$ has Newton polygon vertices $(0, 2)$ and $(3, 0)$; the diagonal $p = q$ meets the edge at $p = 6/5$, so $c_0(f) = 5/6$ (Collins, 2017).
• Generalized cusp: $f(x, y) = y^N + x^M$ with Newton polyhedron vertices $(0, N)$ and $(M, 0)$. The RLCT is $(M + N)/(M N)$ (Collins, 2017).
• Monomial multi-ideal: For ideals $(x)$ and $(y)$ with exponents $e = (e_1, e_2)$, one finds

$$\operatorname{lct}_0(\mathbb{A}^2, (x)^{e_1} (y)^{e_2}) = \min\left(\frac{1}{e_1}, \frac{1}{e_2}, \frac{2}{e_1+e_2}\right)$$

(Ishii, 30 Jun 2025).

• Hyperplane arrangement: For $f(x) = \prod_{j=1}^n L_j(x)^{s_j}$, the RLCT is $\min_W \frac{\operatorname{codim}(W)}{s(W)}$ over all intersections $W$ of the arrangement, with explicit computation via the intersection lattice (Kosta et al., 20 Nov 2024).

5. RLCTs in Bayesian Learning and Statistics

In singular learning theory, RLCTs (or learning coefficients) determine the leading asymptotics of the marginal likelihood ("free energy") and generalization error. For Bayesian models, as $n \to \infty$,

$$Z(n) \sim C n^{-\lambda} (\log n)^{m-1}, \qquad F(n) = nS + \lambda \log n - (m-1) \log \log n + O(1)$$

where $\lambda$ is the RLCT and $m$ its multiplicity (Imai, 2019, Kurumadani, 23 Aug 2024). In regular models, $\lambda = d/2$ and $m=1$, but in singular models these can be fractional or more complex.

For general models, after resolution $K(\theta) \sim a(\theta) \prod u_i^{k_i}$ and the RLCT is the minimum $\frac{h_i + 1}{k_i}$ over charts and directions (Kurumadani, 23 Aug 2024, Imai, 2019).

In applications:

• Mixture models: RLCTs govern the learning rate for mixtures of binomials, with explicit upper bounds given by local computations at non-singular points (Kurumadani, 23 Aug 2024).
• Hyperplane arrangements: Closed combinatorial formulas allow direct computation of RLCTs and their multiplicities, which enter the leading order of marginal likelihood and selection criteria such as sBIC (Kosta et al., 20 Nov 2024).

Recent methods allow RLCT estimation via MCMC using thermodynamic integration and variance identities, enabling model selection in practice without explicit algebraic computation (Imai, 2019).

6. Real and Complex RLCTs: Comparison and Open Problems

In general, $$\operatorname{lct}_{\mathbb{R}}(f) \geq \operatorname{lct}_{\mathbb{C}}(f)$$, reflecting potentially stronger integrability on the real locus. Equality may fail if the real vanishing locus is thinner due to sign oscillations or real structure (Mustata, 2011, Kosta et al., 20 Nov 2024). For hyperplane arrangements where the resolution is defined over $\mathbb{R}$ and all divisors meet the real locus, the real and complex RLCTs agree.

A number of structural questions remain open in the real-analytic category:

• Rationality of RLCTs: In the complex algebraic case, RLCTs are always rational, but in the real-analytic setting, general rationality is conjectural and relies on the existence of sufficiently functorial resolutions and vanishing theorems (Mustata, 2011, Ishii, 30 Jun 2025).
• Openness: It remains open whether the set of $c$ for which $\exp(-2c\varphi)$ is locally integrable at a point is always open, paralleling the complex case Openness Conjecture (Mustata, 2011).
• Multiplier ideals and asymptotics: The structure of real multiplier ideals, jumping numbers, and their analogues to complex-theoretic invariants are only partially understood.

Recent progress has transferred many chain conditions and finiteness results from characteristic zero to positive characteristic via explicit "lifting" constructions, preserving discreteness and ACC for RLCTs in large generality (Ishii, 30 Jun 2025).

7. Algorithmic and Combinatorial Approaches

For certain families of polynomials, notably "sum-of-products" (sop) polynomials, explicit blow-up algorithms and combinatorial proxies are available for computing RLCTs. A nested sequence of blow-up algorithms can resolve singularities of these polynomials to normal crossings in finitely many steps, yielding exact formulas (notably for binomials) and a general simplex linear programming upper bound for the RLCT in the multivariate case (Hirose, 2023).

A table summarizes computational frameworks:

Approach	Scope	Output
Newton polyhedron	Bivariate analytic germs	$1/d_{NP}(f)$
Resolution theory	General analytic/algebraic	$\min_i (b_i+1)/a_i$
Combinatorial (hyperplanes)	Arrangements	$\min_W \operatorname{codim}(W)/s(W)$
LP-based (sop)	Sum-of-products polynomials	Simplex upper bound

These methods have enabled RLCT computations for a range of algebraic, combinatorial, and statistical models, with applications to model selection, learning rates, and asymptotic volume analysis (Kosta et al., 20 Nov 2024, Hirose, 2023).

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For further technical detail on the proofs and extended applications of RLCTs in algebraic and statistical settings, see (Collins, 2017, Ishii, 30 Jun 2025, Kosta et al., 20 Nov 2024, Imai, 2019, Hirose, 2023).